---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '19598'
abstract:
- lang: eng
  text: We establish universal Gaussian fluctuations for the mesoscopic linear eigenvalue
    statistics in the vicinity of the cusp-like singularities of the limiting spectral
    density for Wigner-type random matrices. Prior to this work, the linear eigenvalue
    statistics at the cusp-like singularities were not studied in any ensemble. Our
    analysis covers not only the exact cusps but the entire transitionary regime from
    the square-root singularity at a regular edge through the sharp cusp to the bulk.
    We identify a new one-parameter family of functionals that govern the limiting
    bias and variance, continuously interpolating between the previously known formulas
    in the bulk and at a regular edge. Since cusps are the only possible singularities
    besides the regular edges, our result gives a complete description of the linear
    eigenvalue statistics in all regimes.
acknowledgement: I would like to express my gratitude to László Erdős for his careful
  guidance and supervision of my work. I am also thankful to Jana Reker and Joscha
  Henheik for many helpful discussions. Open access funding provided by Institute
  of Science and Technology (IST Austria).
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Volodymyr
  full_name: Riabov, Volodymyr
  id: 1949f904-edfb-11eb-afb5-e2dfddabb93b
  last_name: Riabov
citation:
  ama: Riabov V. Linear Eigenvalue statistics at the cusp. <i>Probability Theory and
    Related Fields</i>. 2025;193:1183-1237. doi:<a href="https://doi.org/10.1007/s00440-025-01373-w">10.1007/s00440-025-01373-w</a>
  apa: Riabov, V. (2025). Linear Eigenvalue statistics at the cusp. <i>Probability
    Theory and Related Fields</i>. Springer Nature. <a href="https://doi.org/10.1007/s00440-025-01373-w">https://doi.org/10.1007/s00440-025-01373-w</a>
  chicago: Riabov, Volodymyr. “Linear Eigenvalue Statistics at the Cusp.” <i>Probability
    Theory and Related Fields</i>. Springer Nature, 2025. <a href="https://doi.org/10.1007/s00440-025-01373-w">https://doi.org/10.1007/s00440-025-01373-w</a>.
  ieee: V. Riabov, “Linear Eigenvalue statistics at the cusp,” <i>Probability Theory
    and Related Fields</i>, vol. 193. Springer Nature, pp. 1183–1237, 2025.
  ista: Riabov V. 2025. Linear Eigenvalue statistics at the cusp. Probability Theory
    and Related Fields. 193, 1183–1237.
  mla: Riabov, Volodymyr. “Linear Eigenvalue Statistics at the Cusp.” <i>Probability
    Theory and Related Fields</i>, vol. 193, Springer Nature, 2025, pp. 1183–237,
    doi:<a href="https://doi.org/10.1007/s00440-025-01373-w">10.1007/s00440-025-01373-w</a>.
  short: V. Riabov, Probability Theory and Related Fields 193 (2025) 1183–1237.
corr_author: '1'
date_created: 2025-04-20T22:01:28Z
date_published: 2025-12-01T00:00:00Z
date_updated: 2026-04-07T12:32:19Z
day: '01'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.1007/s00440-025-01373-w
external_id:
  arxiv:
  - '2307.07432'
  isi:
  - '001466997300001'
file:
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  creator: dernst
  date_created: 2025-12-30T13:10:05Z
  date_updated: 2025-12-30T13:10:05Z
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has_accepted_license: '1'
intvolume: '       193'
isi: 1
language:
- iso: eng
month: '12'
oa: 1
oa_version: Published Version
page: 1183-1237
publication: Probability Theory and Related Fields
publication_identifier:
  eissn:
  - 1432-2064
  issn:
  - 0178-8051
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
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    relation: dissertation_contains
    status: public
scopus_import: '1'
status: public
title: Linear Eigenvalue statistics at the cusp
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 193
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